Semi-Analytical Bifurcation Trees of Periodic Motions in a Periodically Forced Third-Order Quadratic Nonlinear System

This paper presents semi-analytical periodic motions and bifurcations in a periodically forced third-order quadratic nonlinear system through the implicit mapping method. The third-order quadratic nonlinear system is one of the simplest three-dimensional nonlinear systems, which is used to discuss the global behaviors of periodic motions. From the implicit mapping method, the continuous-time flow is discretized to an implicit point-to-point mapping first, the corresponding periodic motions are obtained through mapping structures, and their stability and bifurcation conditions are determined through eigenvalue analysis. The bifurcation trees of period-1 motion to chaos and the corresponding parameter maps provide the global view of periodic motions in such a third-order quadratic nonlinear system. Numerical simulations of periodic motions on the bifurcation trees of periodic motions are presented. The numerical results demonstrate motion complexity from simple period-1 to period-8 motions. This paper shows simplest third-order quadratic nonlinear systems possess complex motions, and the method presented herein can be used for other nonlinear dynamical systems.